RIS-Enhanced Offset Gregorian Reflector

• The offset Gregorian reflector system is defined by two decentered aspheric mirrors that eliminate central blockage and cancel primary aberrations for clear imaging.
• RIS integration on the subreflector enables real-time sidelobe nulling with < -40 dB null depth while incurring only a 0.2 dB main-lobe directivity penalty.
• Precise mirror geometry and an efficient greedy optimization algorithm ensure high Strehl ratios and diffraction-limited performance in radio astronomy and THz imaging.

An offset Gregorian reflector system is a two-mirror, obstruction-free imaging or antenna architecture characterized by aspheric mirrors (a paraboloidal or ellipsoidal main and an ellipsoidal subreflector), arranged such that their axes are decentered and tilted to eliminate blockage, cancel primary aberrations, and provide high-fidelity far-field patterns. Recent advances integrate reconfigurable intelligent surfaces (RIS) onto the subreflector, enabling real-time sidelobe control with negligible main-lobe directivity penalty, a paradigm now adopted in high-sensitivity radio astronomy and THz imaging.

1. Geometric Configuration and Axis Decentering

An offset Gregorian system follows rigorous geometric definitions and coordinate conventions, central to both optical and RF performance optimization (Ellingson et al., 29 Dec 2025, Ghamsari et al., 2022, Muslimov et al., 2018). The boresight is aligned with the global +z-axis, with both main reflector and subreflector as off-axis sections of parent conic surfaces. The feed sits along the –z-axis behind the subreflector, and the system's chief-ray after M1 emerges on the z-axis; this configuration precludes central obscuration and specular ghosts.

Key parameters for a large radio astronomy implementation (following (Ellingson et al., 29 Dec 2025), Section II) are:

• Main reflector: paraboloid, diameter $D=20$ m, focal length $F=16.56$ m, offset $d_0=11.74$ m; surface described by

$$r_1(\theta, \phi) = \frac{2F \cos\theta}{1+\cos\theta}$$

• Subreflector: ellipsoid of eccentricity $e=0.49$, half-focal distance $c=2.678$ m, rim half-angle $\theta_e=11.95^\circ$, locally parameterized as

$\frac{x'^2}{b^2} + \frac{(z'-z_0)^2}{a^2} = 1$

with rotation angles $\alpha=-15.87^\circ$ (about $x$), $F=16.56$0 (about $F=16.56$1).

In THz imaging and precision optical designs, decenter and tilt enforce the Dragone condition:

$F=16.56$2

where $F=16.56$3 is secondary magnification, guaranteeing first-order coma and astigmatism cancelation (Ghamsari et al., 2022). Mirror positions are determined by tilt $F=16.56$4 (main) and $F=16.56$5 (secondary), clear aperture diameter, and focal distances—all critical inputs for both electromagnetic and geometric ray tracing.

2. Mirror Surface Equations and Aberration Control

Reflector surfaces are defined via rotationally symmetric sag equations (in local coordinates):

$F=16.56$6

where $F=16.56$7 is the radius of curvature and $F=16.56$8 the conic constant ($F=16.56$9 for spheres, $d_0=11.74$0 for paraboloids, $d_0=11.74$1 for ellipsoids, $d_0=11.74$2 for hyperboloids) (Muslimov et al., 2018, Ghamsari et al., 2022).

Example mirror parameters for CEGRS optimization (Ghamsari et al., 2022):

Parameter	Main (M1)	Subreflector (M2)
Conic constant $d_0=11.74$3	–0.852	–2.00
Radius of curvature	1.924 m	0.450 m
Aperture diameter	0.500 m	0.120 m
Tilt angle (y-axis)	15°	7°

Aberration control is achieved via surface geometry and positioning. Primary aberrations follow [Dragone, IEEE TAP 1982]:

• Spherical: $d_0=11.74$4
• Coma: $d_0=11.74$5
• Astigmatism: $d_0=11.74$6

Offset decentering, precise tilts, and appropriate conic selections ensure Strehl ratio $d_0=11.74$7 at 1 THz for RMS wavefront error $d_0=11.74$8m over a $d_0=11.74$9 field (Ghamsari et al., 2022). Curved detectors may be introduced to cancel Petzval curvature and further suppress field-dependent spot broadening (Muslimov et al., 2018).

3. RIS-Equipped Subreflector for Sidelobe Nulling

Offset Gregorian architectures are now integrated with reconfigurable intelligent surfaces (RIS), particularly for radio astronomy interference mitigation (Ellingson et al., 29 Dec 2025). In this approach, the outer one-third of the subreflector is replaced by a passive reflectarray shell consisting of $$r_1(\theta, \phi) = \frac{2F \cos\theta}{1+\cos\theta}$$0 elements (square $$r_1(\theta, \phi) = \frac{2F \cos\theta}{1+\cos\theta}$$1, optimally located for angular sampling). Each patch realizes $$r_1(\theta, \phi) = \frac{2F \cos\theta}{1+\cos\theta}$$2-bit phase-only control:

$$r_1(\theta, \phi) = \frac{2F \cos\theta}{1+\cos\theta}$$3

Or, equivalently, binary scattering coefficients $$r_1(\theta, \phi) = \frac{2F \cos\theta}{1+\cos\theta}$$4.

Electromagnetic scattering follows physical optics (PO), with the H-plane far-field given by:

$$r_1(\theta, \phi) = \frac{2F \cos\theta}{1+\cos\theta}$$5

where $$r_1(\theta, \phi) = \frac{2F \cos\theta}{1+\cos\theta}$$6 is the quiescent field (unmodified system) and $$r_1(\theta, \phi) = \frac{2F \cos\theta}{1+\cos\theta}$$7 is the contribution from patch $$r_1(\theta, \phi) = \frac{2F \cos\theta}{1+\cos\theta}$$8.

Sidelobe-level (SLL) and null depth (ND) metrics are defined as:

$$r_1(\theta, \phi) = \frac{2F \cos\theta}{1+\cos\theta}$$9

$e=0.49$0

For the 20 m/3.2 m system at 1.5 GHz, RIS actuated sidelobe nulling (at $e=0.49$1) achieves $e=0.49$2 dB null depth, with only $e=0.49$3 dB main-lobe directivity loss ($e=0.49$4 dBi, $e=0.49$5 dBi), and only 7 of 282 elements require flipping—demonstrated in (Ellingson et al., 29 Dec 2025), Section V.

4. Optimization Algorithm for Sidelobe Nulling

Efficient RIS state-setting is achieved via an unconstrained greedy algorithm [(Ellingson et al., 29 Dec 2025), Section IV]:

1. Compute $e=0.49$6 and $e=0.49$7 for all $e=0.49$8 at target null direction $e=0.49$9 ($c=2.678$0).
2. Rank elements by $c=2.678$1, yielding permutation $c=2.678$2.
3. Initialize $c=2.678$3.
4. For $c=2.678$4 to $c=2.678$5: a. Set $c=2.678$6 tentatively; if $c=2.678$7, flip $c=2.678$8. b. Update $c=2.678$9
5. Terminate.

This algorithm implicitly minimizes $\theta_e=11.95^\circ$0 without imposing main-lobe constraints and converges monotonically in a single pass, with computational complexity $\theta_e=11.95^\circ$1 plus field evaluations. The empirical penalty on directivity is $\theta_e=11.95^\circ$2 dB (Ellingson et al., 29 Dec 2025).

5. Implementation Constraints and Hardware Integration

RIS subreflectors are realized as deformable, passive reflectarray shells with low-power biasing, e.g., PIN-diode loaded patches controlled via feed support wiring [(Ellingson et al., 29 Dec 2025), Section VI]. Only the outer one-third of the subreflector is actively reconfigurable; mechanical and thermal properties must replicate the conventional subreflector for effective retrofit.

Limitations include:

• Nulling confined to close-in sidelobes (RIS rim zone)
• Integration must preserve global optical alignment tolerances (e.g., surface figure $\theta_e=11.95^\circ$3 at 633 nm, decenter $\theta_e=11.95^\circ$4m, tilt $\theta_e=11.95^\circ$5 in optical systems (Muslimov et al., 2018))
• Retrofits feasible for large research facilities (GBT, ATA, MeerKAT), with mounting options for outrigger RIS panels or full subreflector replacement.

6. Applications in Astronomy and Imaging

Offset Gregorian systems with optimized aberration control and RIS-enabled sidelobe mitigation are widely adopted in radio astronomy, THz imaging, and unobscured astronomical telescopes (Ellingson et al., 29 Dec 2025, Ghamsari et al., 2022, Muslimov et al., 2018).

Key metrics for a radio astronomy RIS system:

• Deep null ($\theta_e=11.95^\circ$6 dB attenuation) suppressing satellite RFI entering via sidelobes near $\theta_e=11.95^\circ$7
• Negligible system temperature impact from $\theta_e=11.95^\circ$8 dB directivity reduction, preserving sensitivity in protected bands
• Compatibility with existing telescope platforms via subreflector swap or modular RIS outrigger attachment

In optical THz imaging, an F-number of 2.47 offers uniform beam quality, spot diameter $\theta_e=11.95^\circ$9 mm at target plane ($\frac{x'^2}{b^2} + \frac{(z'-z_0)^2}{a^2} = 1$0 m), and Strehl ratio $\frac{x'^2}{b^2} + \frac{(z'-z_0)^2}{a^2} = 1$1 over a $\frac{x'^2}{b^2} + \frac{(z'-z_0)^2}{a^2} = 1$2 field (Ghamsari et al., 2022). Unobscured Gregorian layouts with curved detectors achieve nearly diffraction-limited imaging ($\frac{x'^2}{b^2} + \frac{(z'-z_0)^2}{a^2} = 1$3 waves rms across $\frac{x'^2}{b^2} + \frac{(z'-z_0)^2}{a^2} = 1$4 field) (Muslimov et al., 2018).

7. Performance Metrics and Limitations

Performance outcomes are systematically quantified:

• Main-lobe directivity pre- and post-RIS: $\frac{x'^2}{b^2} + \frac{(z'-z_0)^2}{a^2} = 1$5 dBi; $\frac{x'^2}{b^2} + \frac{(z'-z_0)^2}{a^2} = 1$6 dBi (Ellingson et al., 29 Dec 2025)
• Sidelobe peak reduction: $\frac{x'^2}{b^2} + \frac{(z'-z_0)^2}{a^2} = 1$7 dB (quiescent) to $\frac{x'^2}{b^2} + \frac{(z'-z_0)^2}{a^2} = 1$8 dB (RIS-nulled) at $\frac{x'^2}{b^2} + \frac{(z'-z_0)^2}{a^2} = 1$9
• THz imaging: RMS wavefront error $\alpha=-15.87^\circ$0m across $\alpha=-15.87^\circ$1 field, Strehl ratio $\alpha=-15.87^\circ$2, beam efficiency $\alpha=-15.87^\circ$3 (Ghamsari et al., 2022)
• Diffraction-limited spot diagrams: RMS radius $\alpha=-15.87^\circ$4–$\alpha=-15.87^\circ$5m on curved focal plane (Muslimov et al., 2018)

This suggests offset Gregorian reflector systems augmented by RIS or other computational subreflectors represent a convergence of high-performance imaging and electromagnetic control. A plausible implication is a further evolution toward dynamic, full-aperture reconfiguration for broader field sidelobe nulling once hardware integration and active matrix addressing mature.

Table: RIS-Augmented Offset Gregorian Metrics (Radio Astronomy, (Ellingson et al., 29 Dec 2025))

Metric	Quiescent Value	RIS-Optimized Value
Main-lobe directivity (dBi)	48.5	48.3
2nd sidelobe peak (dB)	–17	<–40
Flipped RIS elements	0	7 of 282
Directivity penalty (dB)	—	0.2

These empirical results demonstrate the practical feasibility of RIS-modified offset Gregorian reflectors for advanced radio frequency and optical imaging system applications.